Topic 3: Angle measurement traversing

Aims -Learn what control surveys are and why these are an essential part of surveying -Understand rectangular and polar co-ordinates and how to transform between the two -Learn how to carry out a traverse

Control Surveys All measurements taken for engineering surveys are based on a network of horizontal and vertical reference points called control points. These networks are used on site in the preparation of maps and plans, they are required for dimensional control (setting out) and are essential in deformation monitoring. Because all surveying needs control points at the start of any engineering or construction project a control survey must be carried out in which the positions of the control points are established. The positions of horizontal control are usually specified in rectangular (x and y) coordinates.

Rectangular co-ordinates Any point P, has the coordinates known as easting E p and northing N p quoted in the order E p, N p.

Calculation of rectangular co-ordinates On a coordinate grid, the direction of a line between two points is know as its bearing. The whole circle bearing of a line is measured in a clockwise direction in the range 0 to 360.

The following figure shows the plan of two points A and B on a rectangular grid. If the coordinates of A(E A, N A ) are known, the coordinates of B (E B, N B ) are obtained as follows: Where: EAB = the eastings difference from A to B NAB = the northings difference from A to B DAB = the horizontal distance from A to B θab = the whole circle bearing from A to B

Example: The coordinates of point A are 311.617mE, 447.245mN. Calculate the coordinates of point B, where D AB = 57.916M and θ AB = 37 11 20 and point C where D AC = 85.071m and θ AC = 205 33 55 Polar co-ordinates Another coordinate system used in surveying is the polar coordinate system. Here a point B is located with reference to point A by a polar coordinates D and θ.

D is the horizontal distance from A to B and θ is the whole circle bearing of the line A to B. For the reverse of the previous example where the coordinates are know for both points it is possible to compute the whole circle bearing and horizontal distance of the line between the two points. This is known as rectangular to polar coordinate conversion. Example: The coordinates of A and B are EA = 469.721m, N A = 338.466m and E B = 501.035m, N B = 310.617m. Calculate the horizontal distance D AB and the whole circle bearing θ AB. D AB = E 2 AB + N 2 AB = ( E B E A ) 2 + ( N B N A ) 2 = (501.035 469.721) 2 + (310.617 338.446) 2 = 41.906m

To calculate θ AB a sketch of the line AB must be made in order to identify which quadrant the angle is in (as different equations apply for each quadrant):

θ AB = tan 1 E N AB AB + 180 o = tan 1 31.314 + 180 27.849 o = 131 38 53 Traversing A traverse is a means of providing horizontal control in which the rectangular coordinates of a series of control points located are a site are determined from a combination of angle and distance measurements.

Each point on a traverse is called a traverse station and these must first be located well and marked with ground markers before surveying commences:

Procedure for Traversing When angle ABC is measured: At A a tripod target is set up centred and levelled, at B a theodolite or total station is set up, levelled and centred as normal. At C another tripod target is set up as for A. This enables the horizontal angle at B to be recorded and if a total station is being used the distances BA and BC can be measured. When the angle BCD is measured: At A the tripod target are moved to D, where the target is centred and levelled as before. At B the total station or theodolite is unclamped and interchanged with the target at C (the tripods can remain in the same place and there is no need to recentre them). The horizontal angle at C can now be measured along with the distances CB and CD. The distance CB will provide a check for error in the previous BC measurement.

When the angle CDE is measured: At B the tripod and target are moved to E. The theodolite or total station at C is interchanged with the target at D. The process is repeated for the whole traverse, if 4 or more tripods are used this speeds up the process. Traverse Calculations Traverse calculations involve the calculation of the 1) whole circle bearings 2) the coordinate differences and 3) the coordinates of each control point. To illustrate these calculations we will use the traverse ABCEDFA below throughout:

Errors and Misclosure The first part of a traverse calculation is to check that the observed angles sum to their required value. Sum of internal angles = (2n - 4) x 90o Sum of External angles = (2n + 4) x 90o, n is the number of angle measured If on summing these values a misclosure is found, it is divided equally between the station points if it is acceptable. Acceptability of misclosure E for traversing is given by: E = ±KS n Where K is a multiplication factor from 1 to 3 depending on weather conditions. S is the smallest reading interval on the theodolite (e.g. 20, 5 or 1 ) and n is the number of angle measured. Taking our traverse ABCDEFA the misclosure is calculated and redistributed as follows:

Calculation of Whole Circle Bearings To calculate the coordinates of a control point the WCB must be known as we saw earlier. This is done according to the following formulae:

Forward bearing YZ = Back bearing YX + (for the above example) In general Forward bearing = back bearing + left hand angle A forward bearing is a bearing in the direction of the traverse e.g. XY and YZ, a back bearing is a bearing in the opposite direction to the traverse e.g. YX and ZY. Forward and back bearing differ by ±180. The left hand angle is the angle between the bearing lines at a control station that lies to the left of the station relative to the direction of the traverse, i.e. the internal angle for anticlockwise traverses and the external angle for clockwise traverses.

Example: WCB at station A of the ABCDEFA traverse The calculation of WCBs must start with a known bearing or an assumed arbitrary bearing. Here the first bearing AF is known to be 70 00 00. Because the internal angles have been measured the traverse is calculated in an anticlockwise direction and AF is a back bearing.

Forward bearing AB = back bearing AF + adjusted left hand angle at A = 70 00 00 + 115 11 10 = 185 11 10 (WCB at A) Example : WCB at station B of the ABCDEFA traverse

Forward bearing BC = back bearing BA + adjusted left hand angle at B To convert the forward bearing AB into a back bearing BA we add or subtract 180. Back bearing BA = 185 11 10 ±180 = 05 11 10 Forward bearing BX = 05 11 10 + 95o 00 00 = 100 11 10 (WCB at B) The WCBs of all other stations are carried out in a similar manner. To finalise this section of the calculations the final forward FA bearing must equal the first back bearing AF (250 is equivalent to 70 in this case as shown in the next slide).

Calculation of Coordinate Differences The next stage of the traverse calculation is to determine the coordinate differences of the traverse lines E, N. Example: Traverse ABCDEFA, line AB + BC

Error Check! In order to assess the accuracy of the traverse E (should) = 0 and N (should) = 0, since the traverse starts and finishes in the same place. The errors in this summation e E and e N are:

Coordinate Differences : Bowditch Adjustment Method Following calculation of E, N and the misclosure errors ee and en an adjustment of those errors δe, δn must be made. This method is most suitable for traverses carried out using steel tapes Adjustment to E (or N) for a traverse line = δe (or δn) = - ee (or -en) x length of traverse line / total length of traverse

Coordinate Differences : Equal Adjustment Method This adjustment method is most suited for traverses carried out with total stations Calculation of Coordinates Recalling from earlier that the coordinates of a point are calculated as follows for points B and C: EA, NA were given as 350.000mE, 500.000mN EB = EA ± EAB = 350 7.768 = 342.232mE NB = NA ± NAB = 500 85.517 = 414.483mN This process is repeated until point A is rechecked as shown on the next slide:

Further Examples Below the angles and distances for traverse A1234A are shown. The coordinates of A are 642.515mE, 483.980mN. the traverse is oriented to existing control point B (548.005mE, 594.279mN). Calculate the coordinates of stations 1-4.